Variationplates

# Variationplates - Principle of Minimum Total Potential...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Principle of Minimum Total Potential Energy In a linear elastic body, the strain energy stored in the body due to deformation is 1 2 T V U dV σ ε = ∫ (1) and the loss of the potential energy of the applied loads for a conservative system is the negative of the work done by the loads as the structure is deformed. The strain energy for a three- dimensional isotropic medium on the Cartesian coordinate system is ( 29 1 2 x x y y z z xy xy yz yz xz xz U dxdydz σ ε σ ε σ ε τ γ τ γ τ γ = + + + + + ∫∫∫ (2) Invoking the plane stress simplification adopted for a thin plate, results in ignoring , , yx xz γ γ and z σ . The constitutive law of a two-dimensional medium is ( 29 ( 29 ( 29 2 2 1 1 2 1 x x y y y x xy xy E E E σ ε με μ σ ε με μ τ γ μ = +- = +- = + (3) The displacement components at any pointing the plate, , , , u v w may be represented in terms of the corresponding middle-plane displacement components, , , , u v w by , , x y u u zw v v zw w w =- =- = (4) The force-deformation relations (or kinematic relations) at any point in the plate are ( 29 ( 29 2 , , 2 , , , , , , / 2 / 2 x x x y y y xy y x x y u w u w u v w w ε ε γ = + = + = + + (5) 1 The force-deformation relations (or kinematic relations) at any point on the plate middle plane are ( 29 ( 29 2 , , 2 , , , , , , / 2 / 2 x x x y y y xy y x x y u w u w u v w w ε ε γ = + = + = + + (6) The total strains represented with a bar are sum of the membrane strain and bending strain such that , , , 2 x x xx y y yy xy xy xy zw zw zw ε ε ε ε γ γ = + = + = + (7) Substituting Eqs. (3) into Eq. (2), gives ( 29 2 2 2 2 1 2 2 2 1 x y x y xy E U dxdydz μ ε ε με ε γ μ- = + + + - ∫∫ ∫ (8) Introducing Eqs. (7) and integrating with respect to z leads to the relations m b U U U = + (9) where 2 2 2 1 2 2 2 m x y x y xy C U dxdy μ ε ε με ε γ- = + + + ∫∫ (10) and ( 29 ( 29 2 2 2 , , , , , 2 2 1 2 b xx yy xx yy xy D U w w w w w dxdy μ μ = + + +- ∫∫ (11) where ( 29 3 2 2 and...
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

Variationplates - Principle of Minimum Total Potential...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online